Scale Physics of the Single Particle Schrödinger Equation with Gravitational Self - Interaction

We consider the modification of a single particle Schrödinger equation by the inclusion of an additional gravitational self-potential term which follows from the prescription that the’ mass-density’that enters this term is given by m|ψ(~r, t)|2, where ψ(~r, t) is the wavefunction andm is the mass of the particle. This leads to a nonlinear equation, the ’ Newton Schrodinger’ equation, which has been found to possess stationary self-bound solutions, whose energy can be determined exactly using an asymptotic method. We find that such a particle strongly violates superposition and becomes a black hole as its mass approaches the Planck mass.